Growth and shortening of microtubules: a two-state model approach

In this study, a two-state mechanochemical model is presented to describe the dynamic instability of microtubules (MTs) in cells. The MTs switches between two states, assembly state and disassembly state. In assembly state, the growth of MTs includes two processes: free GTP-tubulin binding to the tip of protofilament (PF) and conformation change of PF, during which the first tubulin unit which curls outwards is rearranged into MT surface using the energy released from the hydrolysis of GTP in the penultimate tubulin unit. In disassembly state, the shortening of MTs includes also two processes, the release of GDP-tibulin from the tip of PF and one new tubulin unit curls out of the MT surface. Switches between these two states, which are usually called rescue and catastrophe, happen stochastically with external force dependent rates. Using this two-state model with parameters obtained by fitting the recent experimental data, detailed properties of MT growth are obtained, we find that MT is mainly in assembly state, its mean growth velocity increases with external force and GTP-tubulin concentration, MT will shorten in average without external force. To know more about the external force and GTP-tubulin concentration dependent properties of MT growth, and for the sake of the future experimental verification of this two-state model, eleven {\it critical forces} are defined and numerically discussed.

In eukaryotic cells, microtubules (MTs) 2 serve as tracks for motor proteins (1)(2)(3)(4)(5)(6), give shape to cells, and form rigid cores of organelles (7)(8)(9)(10). They also play essential roles in the chromosome segregation (11)(12)(13)(14)(15)(16)(17)(18). During cell division, MTs in spindles constantly grow and shorten by addition and loss of the enzyme tubulin (GTPase) from their tips. The attached duplicated chromosomes are stretched apart (through two kinetochores) from one another by the opposing forces (produced by MTs based on different spindles). Recently, many theoretical models have been designed to understand the roles of MTs during chromosome segregation (19 -25). One essential point, to understand how MTs help chromosome segregation during cell division, is to know the mechanism of MT growth and shortening. In this study, inspired by the mechanochemical model for molecular motors (26) and the GTP-cap model and catch bonds model for MT (8,27), a two-state mechanochemical model will be presented.
Electron microscopy indicates that an MT is composed of n parallel protofilaments (PFs, usually 12 Յ n Յ 15 and n ϭ 13 is used in this work), which form a hollow cylinder (7,8,28). Each PF is a filament that made of head-to-tail associated ␣␤ heterodimers. At the tip of an MT, PFs curl outward from the MT cylinder surface. The tip might be in a shrinking GDP-cap state or growing GTP-cap state. In contrast to the tip in the shrinking GDP state, the growing GTP tip is fairly straight. In other words, in a GTP-cap state, the angle between the curled out segment of PFs and MT surface is less than that in the GDP state. In this work, I will only consider the growth and shortening of one single PF. I assume that each step of growth and shortening of one PF contributes L (nm) to the growth and shortening of the whole MT. Intuitively, L ϭ L 1 /n with L 1 the length of one ␣␤ heterodimer. In the numerical calculations, I use L ϭ 8 nm/13 Ϸ 0.615 nm (19,29,45).
My two-state mechanochemical model for PF growth and shortening is schematically depicted in Fig. 1a and mathematically described by a two-line Markov chain in Fig. 1b. In this model, a PF stochastically switches between two states: the assembly and disassembly states. During the assembly state, a PF grows through two processes, 1 3 2 (i), the free GTP-tubulin binding process. The rate constant is k 1 and depends on the GTP-tubulin concentration [tubulin], and 2 3 1 (ii), a PF conformation change process. During which, using energy released from GTP hydrolysis, the curled PF segment is straightened with one PF unit (i.e. one ␣␤ heterodimer) rearranged onto the MT surface, i.e. to parallel to the MT axis approximately. During the disassembly state, each step of a PF shortening also includes two processes, (i) 2 4 1, disassociation of GDPtubulin from a PF tip to the environment and (ii) 1 4 2, one new PF unit curls out from the MT surface (during which phosphate is released from the tip of the tubulin unit simultaneously). The communication among curved PF units in the processes 2 3 1 and 1 4 2 might be explained by similar methods as in the Monod-Wyman-Changeux model (30) or the Koshland-Nemethy-Filmer model (31).
The two-state model presented here can be regarded as a generalization of the one employed by Akiyoshi et al. (27) to explain their experimental data, which is depicted in Fig. 2a.
(My corresponding generalized two-state model, including bead detachment from MT, is depicted in Fig. 2b, see "Modified Model according to Experiments, including Bead Detachment from MT" for detailed discussion). The reasons that I prefer to use this generalized model are as follows. The measurements in Refs. 32-34 indicate that the rate of catastrophe, i.e. transition from elongation to shortening, depends on the GTP-tubulin concentration of the solution. However, for the simple model depicted in Fig. 2a, the catastrophe rate k c is independent of GTP-tubulin concentration. (It is biochemically reasonable to assume that the elongation rate k 1 depends on GTP-tubulin concentration,k 1 ϭ k 1 0 [tubulin], but there is no reason to write k c as a function of [tubulin].) The "Results" show that for my generalized model, the catastrophe rate does change with [tubulin] because GTP-tubulin concentration will change the probabilities of a PF being in states 1 and 2 and consequently change the transition rate from assembly state to disassembly state. At the same time, for the simple model depicted in Fig. 2a, the distribution of the catastrophe time is an exponential. However, experimental measurements under a particular situation indicate that this distribution is clearly not an exponential (34). Certainly, from the parameter values listed in Table 3, one may find that the rates k 1 0 and k 3 are much larger than k 2 0 and k 4 0 , so under low external force and high free GTP-tubulin concentration, the model depicted in Fig. 2a is a good approximation of my generalized model depicted in Fig. 2b. It should be pointed out that, although the model presented here looks more complex, there are only two more parameters than the one depicted in Fig. 2a. 3 The two-state mechanochemical model will be presented and theoretically studied under "Model" and then under "Results" based on the model parameters obtained by fitting the experimental data (mainly obtained in Ref. 27), and properties of MT growth and shortening are studied numerically. This includes external force and GTP-tubulin concentration-dependent growth/shortening speeds (i), mean dwell times in assembly and disassembly state (ii), mean growth or shortening length before the bead, which is used in experiments to apply external force, and detachment from MT. To know more properties of MT dynamics, 11 "critical forces" (detailed definitions will be given in "Critical Forces of MT Growth") are also numerically discussed under "Results," followed by concluding remarks.

MODEL
As the schematic depiction in Fig. 1, a PF might be in two states: the assembly (growth) state and the disassembly (shortening) state. Each of the two states includes two substates, denoted by 1, 2 and 1, 2, respectively. Let p 1 , p 2 be the probabilities that a PF is in assembly substates 1 and 2, respectively, and 1 , 2 be the probabilities that the PF is in disassembly substates 1 and 2, then p 1 , p 2 , 1 , 2 are governed by the following master equation, 3 To keep as few parameters as possible in my two-state model, I assume that the bead only can detach from an MT from substates 1 and 1. The reasons are as follows: in assembly state, the experimental data in Refs. 27  ) should be positive because the growth speed V g ϭ k 1 k 2 L/(k 1 ϩ k 2 ) (see Equation 5). Consequently, the probability p 1 ϭk 2 /(k 1 ϩ k 2 ) that an MT in state 1 (see Equation 12) increases, but the probability p 2 ϭk 1 /(k 1 ϩ k 2 ) that an MT in state 2 decreases with external force, i.e. as the increase of external force, the assembly MT would more like to stay in state 1. Furthermore, from the experimental data in Ref. 27, one sees the detachment rate from the assembly state increases with external force. Therefore, the more reasonable choice is to assume that the bead can only detach from state 1 but not state 2. Through similar discussion, one also can see that it is more reasonable to assume that in disassembly state the bead can only detach from state 1. At the same time, the experimental data in Ref. 32 imply the catastrophe rate decreases with GTP-tubulin concentration [tubulin] (or see Fig. 4b). Because k 1 ϭ k 1 0 ͓tubulin͔, the probability p 1 ϭ k 2 /(k 1 ϩ k 2 ) decreases with [tubulin], but the probability p 2 ϭ k 1 /(k 1 ϩ k 2 ) increases with [tubulin]. This is why I assume the catastrophe takes place at state 1. In assembly state, the growth of a PF is accomplished by two processes, one GTP-tubulin binds to the tip of PF (with GTP-tubulin concentration dependent rate k 1 ), and one PF unit rearranges onto the MT surface (with external force dependent rate k 2 ). The energy used in the second process comes from the GTP hydrolysis in the penultimate tubulin unit. One tubulin unit binding to the tip of PF is assumed to be equivalent to L (nm) growth of the whole MT (L ϭ 0.615 nm is used in this work (19,29)). Similarly, in the disassembly state, the shortening of PF also includes two processes, one PF unit detaches from the tip of PF and one new PF unit curls out the MT surface. In this depiction, the same described previously (25), a segment five dimers in length is assumed to curls out from the MT surface. dp 1 /dt ϭ k 2 p 2 Ϫ k 1 p 1 ϩ k r 1 Ϫ k c p 1 , dp 2 /dt ϭ k 1 p 1 Ϫ k 2 p 2 , where k 1 is the rate of GTP-tubulin binding to the tip of a PF, k 2 is the rate of PF realignment with one new unit lying in the MT surface, k 3 is the dissociation rate of GDP-tubulin from the tip of a PF, and k 4 is the rate of curling out of one tubulin unit from the MT surface (with P i release simultaneously). The steady state solution of Equation 1 is as shown in Equation 2.
One can easily show that the mean steady state velocity of MT growth or shortening is as shown in Equation 3 (35,36), where L is the effective step size of MT growth corresponding to one step of PF growth, and V Ͻ 0 means the MT is shortening with speed ϪV on average. Let p 1 , p 2 be the probabilities that a PF is in substates 1 and 2, respectively, provided the PF is in assembly state, then p 1 , p 2 satisfy that shown in Equation 4.
One can easily show that at steady state, the mean growth speed of a MT with a PF in assembly state is as shown in Equation 5.
Similarly, the mean shortening speed of a MT with a PF in disassembly state is shown in Equation 6.

Modified Model according to Experiments (including Bead Detachment from MT)
To know the model parameters k i , i ϭ 1, …, 4 and k c , k r , I need to fit the model to experimental data. In recent experiments, Akiyoshi et al. (27) attached a bead prepared with kinetochore particles to the growing end of MTs, and constant tension was applied to the bead using a servo-controlled laser trap. In their experiments, the force-dependent mean growth and shortening speeds of MTs were measured. Meanwhile, the rates of rescue and catastrophe, the force-dependent mean lifetime, during which the bead remains attached to MT, and mean detachment rates of the bead during assembly and disassembly states are also measured. Therefore, to fit these experimental data, the model depicted in Fig. 1 should be modified to include the bead detachment processes (see Fig. 2b).
For the model depicted in Fig. 2b, the formulations of mean growth velocity V and mean growth and shortening speeds V g and V s are the same as in Equations 3, 5, and 6. In the following, I will derive an expression of the mean lifetime of the bead on MTs. Let T 1 , T 2 , T 1Ј , T 2Ј be the mean first passage times (MFPTs) of a bead to detachment, when the initial substates are 1, 2, 1, 2, respectively, and then T 1 , T 2 , T 1Ј , T 2Ј satisfy (37-39) as follows.
Then, the mean lifetime is as shown in Equation 8, where p 1 , p 2 , 1 , and 2 can be obtained by the formulation in Equation 2 .  (27) in which both the assembly and disassembly of a PF are assumed to include only one process, described by rates k 1 and k 2 , respectively. b, modified mechanochemical model with bead detachment. In the experiments of Ref. 27, Akiyoshi et al. attached a bead prepared with kinetochore to the growing end of MTs and measured not only the force-dependent mean growing and shortening speeds and switch rates between assembly and disassembly states (i.e. rates of rescue and catastrophe) but also the mean lifetime of the bead on MTs and the rates of bead detachment during assembly and disassembly states. Therefore, to obtain the model parameters and discover more properties of MT growth and shortening, this modified model is used in this study. The main difference between these two models is that the rate of detachment from the assembly state and the rate of catastrophe in model b depend on GTP-tubulin concentration, but they do not in model a.
In assembly state, let T a1 and T a2 be the MFPTs to detachment of the bead initially at substates 1 and 2 respectively, and then T a1 and T a2 satisfy Equation 9.
One can easily show the following (Equation 10).
Therefore, the MFPT to detachment of the bead in assembly state is as shown in Equation 11, where the steady state probabilities of Equation 12 are obtained from Equation 4.
The mean detachment rate during assembly can then be obtained as K a ϭ 1/T a ϭ 1/(p 1 T a1 ϩ p 2 T a2 ), i.e. as shown in Equation 13.
Let T c1 and T c2 be the MFPTs of an MT to catastrophe from substates 1 and 2, respectively, and then T c1 and T c2 satisfy (see Fig. 2b).
The mean rate of catastrophe is then K c ϭ 1/T c ϭ 1/(p 1 T c1 ϩ p 2 T c2 ). The explicit expression can be obtained by replacing k a with k c in Equation 13.
Similarly, the mean rate of rescue is as follows.

Force and GTP-Tubulin Concentration Dependence of Transition Rates
From the experimental data in Ref. 27 (or see Fig. 3), one sees that some transition rates in my model should depend on the external force. Because the processes 1 3 2 and 2 4 1 are accomplished by binding a tubulin unit to, and releasing tubulin unit from, the tip of a PF (see Figs. 1 and 2b), I assume that k 1 and k 3 are force-independent. Similar to the methods demonstrated in the models of molecular motors (26,40) and models for adhesive of cells to cells (41), the external force (F) dependence of rates k 2 , k 4 , k a , k d , k r , k c are assumed to be as follows.
Henceforth, the external force F is positive if it points in the direction of MT growth. Meanwhile, the rate k 1 should depend on the concentration of free GTP-tubulin in solution. Similar to the method in Ref. 26, I simply assume k 1 ϭ k 1 0 ͓tubulin͔.

Critical Forces of MT Growth
For a better understanding of the external force-dependent properties of MTs and the experimental verification of the twostate model, in the following, I will define altogether 11 critical forces. Corresponding numerical results will be presented in the next section.
at F c1 , the average speeds of assembly and disassembly are the same. From Equations 5 and 6, one sees that F c1 satisfies the following.
2. Critical Force F c2 -At this critical value of force, the mean velocity of MT growth is zero. Formulation (3) gives 3. Critical Force F c3 -At this critical value of force, p 1 ϩ p 2 ϭ 1 ϩ 2 , i.e. the probabilities that MTs in assembly and disassembly states are the same. From expressions in Equation 2, one easily sees F c3 that satisfies the following.
4. Critical Force F c4 -At this critical value of force, the detachment rates during assembly and disassembly states are the same. In view of Equations 13 and 14, one can get F c4 by K a (F c4 ) ϭ K d (F c4 ).
5. Critical Force F c5 -At this critical value of force, the mean dwell times of an MT in assembly and disassembly states are the same.
Let T g1 and T g2 be the MFPTs of the bead to detachment or catastrophe of the MT with initial substates 1 and 2, respectively. Then, T g1 , T g2 satisfy (see Fig. 2b and Refs. 37 and 38).
The solutions are as follows.
The mean dwell time of an MT in assembly (or growth) state is then as follows.
Similarly, the mean dwell time of an MT in disassembly (or shortening) state can be obtained as follows.
The critical force F c5 can then be obtained by T g (F c5 ) ϭ T s (F c5 ). 6. Critical Force F c6 -At this critical value of force, the mean lifetime of the bead on MT attains its maximum, i.e. T(F c6 ) ϭ max F T(F) with T given by formulation 8.
7. Critical Force F c7 -At this critical value of force, the mean growth length of MT attains its maximum. The mean growth length of an MT can be obtained by l ϩ ϭ VT with V, T satisfy formulations 3 and 8, respectively.
8. Critical Force F c8 -At this critical value of force, the mean shortening length of MT attains its maximum. The mean shortening length of an MT can be obtained by l Ϫ ϭ ϪVT with V, T satisfy formulations 3 and 8, respectively.
9. Critical Force F c9 -The rates of catastrophe and rescue are the same, i.e. K c (F c9 ) ϭ K r (F c9 ) (see Equations 16 and 17). Under critical force F c9 , the average switch time between growth and shortening, i.e. 1/K c and 1/K r , are the same. This indicates that the mean duration for each growth and each shortening period is the same.
10. Critical Force F c10 -At this critical value of force, V g T g ϭ V g T s . Here, V g T g ϭ: l g is the mean growth length before bead detachment or catastrophe, and V s T s ϭ: l g is the mean shortening length before bead detachment or rescue. The formulations of V g , V g and T g , T s are in Equations 5 and 6 and 24 and 25.
11. Critical Force F c11 -At this critical value of force, V g /K c ϭ V s /K r . Here, V g /K c ϭ: l g * is the mean growth length before catastrophe, and V s /K r ϭ l s * is the mean shortening length before rescue. The formulations of K c , K r are in Equations 16 and 17.
It needs to be clarified that the definitions for F c1 , F c2 , F c3 , F c9 , F c11 are unrelated to bead detachment, but the definitions for F c4 , F c5 , F c6 , F c7 , F c8 , F c10 are related. Therefore, the values of F c1 , F c2 , F c3 , F c9 , F c11 obtained in this theoretical study can be verified by various experimental methods as described (13,(32)(33)(34)(42)(43)(44), but the values of F c4 , F c5 , F c6 , F c7 , F c8 , F c10 can only be verified by a similar experimental method to that in Ref. 27. For the sake of convenience, and based on the above definitions and numerical calculations under "Results" (see Figs. 7 and 8), basic properties of the 11 critical forces, F ci , are listed in Table 1. The main symbols used in this study are listed in Table 2.

RESULTS
To discuss the properties of MT growth and shortening, the model parameters, i.e. k 1 0 , k 3 and k i 0 , ␦ i for i ϭ 2, 4, a, d, r, c should be obtained first. By fitting the expressions of V g , V s , T, K a , K d , K c , K r , which are given in Equations 5,6,8,13,14,16, and 17, respectively, to the experimental data (mainly measured in Ref. 27), these parameter values are obtained ( Fig. 3 and Table 3; the fitting methods are discussed. 4 The data corresponding to zero external force in Fig. 3, b and c (two black dots on vertical axis) are obtained by fitting the corresponding measurements in Ref. 32 with a constant (see the two lines in Fig. 4, a and b) because, as implied by my model, the rates of MT shortening and rescue are independent of GTP-tubulin concentration. All the following calculations will be based on the parameters listed in Table 3. The curve in Fig. 4b is the theoretical prediction of GTP-tubulin concentration dependent  Fig. 3a. Finally, all of the parameters were slightly adjusted according to the experimental data about the mean lifetime of bead attachment to MT (see formulation 8 and Fig. 3d). All of the fitting was done by the nonlinear least square program lsqnonlin in Matlab. In each fitting, I randomly chose 1000 initial values of the parameters and adopted the parameter values that fit the experimental data best.

TABLE 1 Basic properties of the critical forces as defined under "Critical Forces of MT Growth"
See also Figs. 7 and 8.
Rate constants (Fig. 2b) Eq. 18 NOVEMBER 11, 2011 • VOLUME 286 • NUMBER 45 catastrophe rate K c by formulation (16). Compared with the experimental data measured in Refs. 32 and 34, these predictions look satisfactory. 5 From Fig. 5a, one can see that the MT is mainly in assembly state. Further calculations indicate that the ratio of probabilities in assembly state to disassembly state, i.e. (p 1 ϩ p 2 )/( 1 ϩ 2 ), increases exponentially with external force F (Fig. 6a). In experiments of Akiyoshi et al. (27), the external force F is applied to the MT through a bead attached to its growing tip. Fig. 5b indicates that for F Յ 16 pN, the mean dwell time of the MT in assembly state before bead detachment is larger than that in disassembly state. Although the MT is mainly in assembly state, its mean growth velocity is negative under a small external force (Fig. 5c) because for such cases, the shortening speed is much larger than the growth speed (see Fig. 3b). But Fig. 5c indicates the mean velocity of MT growth always increases with external force. Similar to the mean growth velocity, the mean growth length of an MT, before bead detachment, might be negative (i.e. MT shortens its length in long time average, see Fig. 5d), although the MT spends most of its time in assembly state (Fig. 5b). Similar to the mean lifetime (Fig. 3d), the mean growth length of an MT also has a global maximum for external force (Fig. 5d). As mentioned under the "Introduction," the chromosome segregation is accomplished by the tensile force generated during the MT disassembly, Fig. 5 tells us the critical force of one MT disassembly is ϳ1.2 pN under the present experimental environment (27). In Fig. 6b, the mean 5 The parameter values listed in Table 3 did not fit well to the GTP-tubulin concentration-dependent growth speed of MTs obtained in Refs. 32 Table 3 also do not fit well to their data. One can verify that the velocity-force data in Ref. 43 can be well described by formulation 5 but with parameters k 1 0 ϭ 2.99 s Ϫ1 M Ϫ1 , k 2 0 ϭ 53.17 s Ϫ1 and ␦ g ϭ 4.13. The difference among these experimental data might be due to differences of experimental techniques, methods, materials, or even temperature.  Table 3. The two black dots on the vertical axis of b and c are obtained by averaging the data in Ref. 32 (see the lines in Fig. 4, a and b).

TABLE 3 Model parameters obtained by fitting the expressions in Equations 5, 6, 8, 13, 14, 16, and 17 to the experimental data mainly measured in Ref. 27
In the fitting, k B T ϭ 4.12 pN⅐nm and effective step size L ϭ 0.615 nm are used (19,29,45). The fitting results are plotted in Fig. 3.

Parameter Value
growth length l g , l g * and mean shortening length l s , l s * , which are given in the definitions of critical force F c10 , F c11 are also plotted as functions of external force. One can easily see that l g Յ l g * , and l s Յ l s * because the mean dwell time of an MT in assembly state T g Յ 1/K c and mean dwell time in disassembly state T s Յ 1/K r . But for a large external force, l s Ϸ l s * because, for such cases, MTs leave disassembly state mainly by rescue.
As the assembly of an MT depends on the free GTP-tubulin concentration (in my model, the simple relation k 1 ϭ k 1 0 ͓tubulin͔ is used, and the disassembly process is assumed to be independent of the GTP-tubulin concentration, which can be verified by the data in Ref. 32 can see that the critical forces F ci for i ϭ 4, 5, 6 increase, but others decrease with [tubulin]. For high GTP-tubulin concentration, [i]F c2 Ϸ F c11 and F c3 Ϸ F c9 because for k c Ͻ Ͻ k 1 , equations V g /K c ϭ V s /K r and K c ϭ K r can be well approximated by k 1 K r ϭ k 3 k c and k r k 4 (k 1 ϩ k 2 ) ϭ k c k 2 (k 3 ϩ k 4 ). 6 Because the force distribution factors ␦ g Ͼ 0, ␦ s Ͻ 0 (see Table 3), from Equations 5 and 6, one can easily show that the growth speed V g increases, but the shortening 6 If the simple model depicted in Fig. 2a is employed to describe the dynamic properties of an MT, then F c2 ϭ F c11 and F c3 ϭ F c9 . The reason is as follows. At steady state, the probabilities p, that an MT in assembly and disassembly states are p ϭ k r /(k c ϩ k r ) and ϭ k c /(k c ϩ k r ), respectively. Thus, the mean growth velocity of the MT is V ϭ k 1 p Ϫ k 2 ϭ (k 1 k r Ϫ k 2 k c )/(k c ϩ k r ). Then, the critical force F c2 satisfies k 1 (F c2 )k r (F c2 ) ϭ k 2 (F c2 )k c (F c2 ). Furthermore, l g * ϭ V g /K c ϭ V g /k c ϭ k 1 L/k c and l s * ϭ V s /K r ϭ V s /k r ϭ k 2 L/k r , so l g * (F c11 ) ϭ l s * (F c11 ) is equivalent to k 1 (F c11 )k r (F c11 ) ϭ k 2 (F c11 )k c (F c11 ) which means F c2 ϭ F c11 . At the same time, p ϭ is equivalent to k c ϭ k r , so F c3 ϭ F c9 . But for my model as depicted in Fig. 2b, F c2 F c11 and F c3 F c9 (see Fig. 8, b and c).  al. (32). a, shortening speed of an MT and their average value. b, switch rates of an MT between assembly and disassembly where the curve is obtained by my theoretical model using the parameter listed in Table 3 (see formulation 16 with k 1 ϭ k 1 0 ͓tubulin͔), the solid squares are experimental data from Ref. 34.  Table 3. a, under external force, the MT is mainly in assembly state, both the probabilities p 1 and p 2 of an MT in assembly substates 1 and 2 increase with force but with p 2 Ͼ Ͼ p 1 . During disassembly state, the MT is mainly in substate 2, 2 Ͼ 1 . Here, p 1 , p 2 , 1 , 2 are calculated by formulations in 2. b, the dwell time T g of MT in assembly state is always larger than that in disassembly state (denoted by T s ) for external forces Ͻ16 pN (see formulations 24 and 25 for T g and T s ). Similar to the mean lifetime of bead attachment to the MT (see Fig. 3d), both T g and T s increase first and then decrease with the external force. c, the mean velocity of MT growth (see formulation 3) increases monotonically with external force where the negative velocity means the MT shortens its length in a long time average, though the curves in a and b imply that the MT is mainly in assembly state. d, the mean growth length before bead detachment increases first and then deceases with the external force. Here, the mean growth length is obtained by the mean growth velocity of an MT multiplied by the mean lifetime of the bead, i.e. VT (see formulation 3 for V and formulation 8 for T). NOVEMBER 11, 2011 • VOLUME 286 • NUMBER 45 Table 1). Equations 5 and 6 also indicate that the growth speed V g increases with [tubulin], but the shortening speed V s is independent of [tubulin]. Therefore, the critical force F c1 decreases with GTP-tubulin concentration [tubulin] (Fig. 8a). But for high [tubulin], critical force F c1 is almost a constant (see Fig. 7a) because, for saturated concentration, the growth speed V g tends to a constant (see Equation 5 and Fig. 9a). The decrease of critical force F c2 with [tubulin] can be seen easily from expression (see Fig.  7b) (20). The decrease of critical forces F c1 , F c2 implies that low GTP-tubulin concentration might be helpful to chromosome segregation. From expressions in Ref. 2, one can verify (p 1 ϩ p 2 )/( 1 ϩ 2 ) ϭ k r k 4 [k 1 ϩ k 2 ]/k c k 2 [k 3 ϩ k 4 ]. So (p 1 ϩ p 2 )/( 1 ϩ 2 ) increases linearly with [tubulin] (Fig. 10a). At the same time, ␦ r ϩ ␦ s Ͼ 0, ␦ g Ͼ 0 and ␦ g ϩ ␦ c Ͻ 0, ␦ s Ͻ 0 (see Table 3) imply (p 1 ϩ p 2 )/( 1 ϩ 2 ) also increases with external force F (see Fig. 6a). Therefore, the critical force F c3 decreases with [tubulin] (see Fig. 7c).

Dynamics of Microtubules
Because the detachment rate K a increases and detachment rate K d decreases, with the external force F (see Fig. 3a), and K a increases with but K d is independent of [tubulin] (see Equations 13 and 14), the critical force F c4 increases with [tubulin] (see Fig.  7a). The increase of critical force F c5 indicates that MTs will spend more time in assembly state at high GTP-tubulin concentration (see Table 1 and Figs. 7a and 5b). The increase of critical force F c6 (see Fig. 8d) implies that the peak of the lifetime-force curve as plotted in Fig. 3d will move toward the right as the increase of [tubulin], but with an upper bound at around 4 pN (see Figs. 7d and 9d). Similarly, the decrease of critical force F c7 (see Fig. 7d) means the peak of the mean growth length-force curve will move leftwards with the increase of [tubulin], and with lower bound at ϳ4.44 pN. Finally, critical forces F c8 , F c9 , F c10 , F c11 all decrease with [tubulin]. I should mention that in Ref. 27 only experimental data for positive force cases are measured, and similar experimental methods as used in FIGURE 6. a, the ratio of the probability p 1 ϩ p 2 that an MT is in assembly state to the probability 1 ϩ 1 that an MT is in disassembly state increases exponentially with the external force, and under a positive external force, the MT mainly stays in the assembly state, although the assembly speed might be much lower compared with the disassembly speed (see Fig. 3b). b, the mean growth length l g , l g * , and shortening length l s , l s * of an MT in one assembly/ disassembly period. The difference between l g , l s and l g * , l s * is that in the calculation of l g * , l s * , the bead attached to the tip of MT, through which the external force is applied to the MT, is assumed to keep attached to the MT, or the external force is just applied by other methods (32)(33)(34)43), so the MT can only leave the assembly state by catastrophe and leave the disassembly state by rescue.  (27). To better understand the curves for F ci , see Table 1.
Refs. 34 and 43 might be employed to apply negative force to MTs. At the same time, the mechanism of MT growth and shortening under negative external force cases might be completely different from that under positive external force cases, so for the results of critical forces plotted in Fig. 7, which have negative values, experimental verification should be first done before further analysis. . GTP-tubulin concentration dependent properties of MT growth and shortening. a, the mean growth velocity V (see formulation 3) increases with GTP-tubulin concentration. The plots also indicate V increases with external force (see Fig. 5c). b and d, the mean lifetime of bead attachment to the MT increases with GTP-tubulin concentration but increases first and then decreases with the external force. For high GTP-tubulin concentration, the critical force F c6 under which the mean lifetime gets its maximum is almost a constant (ϳ4 pN), see also Fig. 7d. c, the mean growth length of an MT before bead detachment does not change monotonically with the external force and GTP-tubulin concentration, so there exists a critical force under which the maximum is obtained. But for high GTP-tubulin concentration, mean growth length increases with [tubulin]. d, for any GTP-tubulin concentration, the mean lifetime does not change monotonically with the external force. The optimal value of external force, under which the mean lifetime is maximum, increases with GTP-tubulin concentration but is almost invariable for large [tubulin]. c and d, both the mean growth length and mean lifetime do not change monotonically with the external force, so there exist optimal values under which the corresponding maximum is reached (see F c6 , F c7 in Fig. 7). .) The mean velocity V and mean growth speed V g also increase with [tubulin] but tend to an external force F-dependent constant (one can verify that this limit constant is k 2 L ϭ k 2 0 L exp(F␦ g /k B T). For such cases, the MT stays mainly in substate 2, i.e. p 2 Ϸ 1, see Fig. 5a). The mean growth length VT does not change monotonically with external force (see Figs. 5a and 9c) but increases with [tubulin] for high GTP-tubulin concentration cases.

CONCLUDING REMARKS
In this work, a two-state mechanochemical model of MT growth and shortening is presented. In assembly (growth) state, one GTP-tubulin will attach to the growing tip of the PF first and then, after the hydrolysis of GTP in the penultimate PF unit, the curved PF segment is slightly straightened with one new PF unit lying in the MT cylinder surface. In the disassembly (shortening) state, one tubulin unit will detach from the tip of PF, and then the GDP (or GDPϩP i ) capped tip segment of PF will be further curved with one new tubulin unit out of the MT surface. (The phosphate is assumed to be released simultaneously.) The PF can switch between the assembly and disassembly states with external force dependent rates stochastically. Each assembly or disassembly process contributes to one step of growth or shortening of MT with step size L ϭ 0.615 nm. This model can fit the recent experimental data measured by Akiyoshi et al. (27) as well.
From this model, interesting properties of MT growth and shortening are found. Under large external force or high GTPtubulin concentration, the MT is mainly in assembly state. The mean lifetime of bead attachment to the MT and mean growth length during this period (in experiments, the external force is applied to MT through a bead attached to the growing tip of MT) increase first and then decrease with the external force. Roughly speaking, they all increase with the GTP-tubulin concentration; the growth speed of an MT increases with GTPtubulin concentration but has an external force-dependent limit. For the sake of experimental verification, altogether 11 critical forces are defined, including the force under which the mean lifetime or mean growth length reach its maximum, the mean assembly speed is equal to the mean disassembly speed, the probabilities of MT in assembly and disassembly states are equal to each other, the detachment rates of bead during assembly and disassembly states are the same, the mean dwell times in assembly and disassembly states are the same, the mean growth velocity of MT vanishes, and etc. Almost all of the above critical forces decrease with the GTP-tubulin concentration, as high GTP-tubulin concentration is favorable for MT growth, and under low GTPtubulin concentration, an MT will shortens its length in average. Roughly speaking, GTP-tubulin and external forces are helpful to MT assembly, but there exists optimal values of the external forces for the mean lifetime of the bead on MT and the mean growth length of MT.